1. Lim - Sweb.cz

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1. lim (x2 − 1) = 22 − 1 = 3 x→2 2. 2·2−3 1 2x − 3 = 2 = 2 x→2 x + 1 2 +1 5 lim 3. √ lim sin x = x→ Π 3 4. 3 2 x(1 + 2x) x + 2x2 = lim = lim (1 + 2x) = 1 + 2 · 0 = 1 x→0 x→0 x→0 x x lim 5. x2 + 3x + 2 −1 + 2 1 (x + 1)(x + 2) x+2 = lim = lim 2 = = x→−1 x→−1 (x + 1)(x2 − x + 1) x→−1 x − x + 1 x3 + 1 (−1)2 − (−1) + 1 3 lim 6. x2 − 9 (x + 3)(x − 3) = lim = lim (x − 3) = −3 − 3 = −6 x→−3 x + 3 x→−3 x→−3 x+3 lim 7. x3 − 1 (x − 1)(x2 + x + 1) = lim = lim (x2 + x + 1) = 12 + 1 + 1 = 3 x→1 x − 1 x→1 x→1 x−1 lim 8. (x − 3)(x − 1) x2 − 4x + 3 = lim = lim (x − 1) = 3 − 1 = 2 x→3 x→3 x→3 x−3 x−3 lim 9. x2 + 5x + 6 1 1 (x + 2)(x + 3) x+3 = lim = lim = =− x→−2 x2 + x − 2 x→−2 (x + 2)(x − 1) x→−2 x − 1 −3 3 lim 10. x3 − 5x2 + 8x − 4 x→2 x3 − 3x2 + 4 Protože (x3 − 5x2 + 8x − 4) : (x − 2) = x2 − 3x + 2 a (x3 − 3x2 + 4) : (x − 2) = x2 − x − 2, máme lim (x − 2)(x2 − 3x + 2) 1 x2 − 3x + 2 (x − 2)(x − 1) x−1 = lim 2 = lim = lim = 2 x→2 (x − 2)(x − x − 2) x→2 x − x − 2 x→2 (x − 2)(x + 1) x→2 x + 1 3 lim 11. 1 3 sin 3x 3 sin 3x 3 3 sin 3x = lim · = lim = ·1= x→0 5 x→0 5x 3x 5 x→0 3x 5 5 lim 12. 2 cos2 x sin x sin x = lim 2 · lim cos2 x · lim = 2 · cos2 0 · 1 = 2 · 1 · 1 = 2 x→0 x→0 x→0 x→0 x x lim 13. 1 − cos x 1 − cos x 1 + cos x 1 − cos2 x sin2 x 1 = lim · = lim 2 = lim · 2 2 x→0 x→0 x x 1 + cos x x→0 x (1 + cos x) x→0 x2 1 + cos x 2  1 1 1 sin x · lim = 12 · = = lim x→0 1 + cos x x→0 x 2 2 lim 1 14. 1 − cos 2x 1 − cos2 x + sin2 x 2 sin2 x 2 sin x = lim = lim = lim = 2·1 = 2 x→0 x→0 x→0 x sin x x→0 x sin x x sin x x lim 15.  lim x→0 3 x − x2 + 1 sin x  = lim x→0 3 x 3 1 1 − lim = 2 − lim = 3− = 2 x2 + 1 x→0 sin x 0 + 1 x→0 sinx x 1 16. 2 tan2 x 2 sin2 x 2 = lim · lim = lim 2 2 x→0 3 cos2 x x→0 x→0 x→0 3x cos2 x 3x  lim sin x x 2 = 2 2 2 ·12 = = 3 cos2 0 3·1 3 17. lim x→0 sin 3x 3 sin 3x x sin 3x = lim · = 3· lim · lim x→0 x→0 tan x 3x tan x 3x x→0 1 tan x x 1 = 3·1· lim cos x· sin x = 3·1 = 3 x→0 x 18. √ √   x+2+1 x+1 x+1 (x + 1)( x + 2 + 1) = lim √ ·√ = lim x→−1 x→−1 x+2−1 x + 2 − 1 x→−1 x+2−1 x+2+1 √ √ √ (x + 1)( x + 2 + 1) = lim = lim ( x + 2 + 1) = −1 + 2 + 1 = 2 x→−1 x→−1 x+1 lim √ 19. √ √ √   2− x−3 2+ x−3 2− x−3 4 − (x − 3) √ √ = lim · lim = lim 2 x→7 x→7 x→7 x2 − 49 x2 − 49 2+ x−3 (x − 49)(2 + x − 3) = lim x→7 7−x −(x − 7) √ √ = lim (x2 − 49)(2 + x − 3) x→7 (x − 7)(x + 7)(2 + x − 3) (−1) −1 −1 1 √ √ = = =− x→7 (x + 7)(2 + 14 · 4 56 x − 3) 14 · (2 + 7 − 3) = lim 20. √ ! √ x+2+ 2 sin 2x sin 2x √ = lim √ √ ·√ √ lim √ x→0 x + 2 − 2 x→0 x+2− 2 x+2+ 2 √ √ √ √ sin 2x · ( x + 2 + 2) 2 · sin 2x · ( x + 2 + 2) = lim = lim x→0 x→0 x+2−2 2x √ √ √ √ √ sin 2x = 2 · lim · lim ( x + 2 + 2) = 2 · 1 · ( 2 + 2) = 4 2 x→0 2x x→0 21. lim (5x3 −x2 +2x+1) = lim 5x3 (1− x→+∞ x→+∞ x2 2x 1 1 2 1 + + ) = lim 5x3 · lim (1− + 2 + 3 ) x→+∞ x→+∞ 5x3 5x3 5x3 5x 5x x = lim 5x3 · 1 = lim 5x3 = +∞ x→+∞ x→+∞ 22. lim (5x3 −x2 +2x+1) = lim 5x3 (1− x→−∞ x→−∞ x2 2x 1 1 2 1 + + ) = lim 5x3 · lim (1− + 2 + 3 ) x→−∞ x→−∞ 5x3 5x3 5x3 5x 5x x = lim 5x3 · 1 = lim 5x3 = −∞ x→−∞ x→−∞ 2 23. lim (−5x3 −x2 +2x+1) = lim −5x3 · lim (1+ 1 1 2 − 2 − 3 ) = lim −5x3 ·1 = −∞ x→+∞ 5x 5x x lim (−5x3 −x2 +2x+1) = lim −5x3 · lim (1+ 1 2 1 − − ) = lim −5x3 ·1 = +∞ x→−∞ 5x 5x2 x3 x→+∞ x→+∞ x→+∞ 24. x→−∞ x→−∞ x→−∞ 25. lim (2x4 −3x3 +x+1) = lim 2x4 (1− 1 1 3 + + ) = lim 2x4 = +∞ x→+∞ 2x 2x3 2x4 lim (2x4 −3x3 +x+1) = lim 2x4 (1− 1 1 3 + 3 + 4 ) = lim 2x4 = +∞ x→−∞ 2x 2x 2x x→+∞ x→+∞ 26. x→−∞ x→−∞ 27. lim (−2x4 −3x3 +x+1) = lim −2x4 (1+ 3 1 1 − − ) = lim −2x4 = −∞ x→+∞ 2x 2x3 2x4 lim (−2x4 −3x3 +x+1) = lim −2x4 (1+ 1 1 3 − − ) = lim −2x4 = −∞ x→−∞ 2x 2x3 2x4 x→+∞ x→+∞ 28. x→−∞ 29. x→−∞ 3 1 x2 · ( x3 − x12 ) 0 3x − 1 x − x2 = lim = =0 = lim x→+∞ 1 + 1 x→+∞ x2 · (1 + 1 ) x→+∞ x2 + 1 1 x x lim 30. x2 (5 − 5x2 − 4x + 2 = lim x→+∞ 3x2 + x − 3 x→+∞ x2 (3 + lim 4 x 1 x + − 2 x2 ) 3 x2 ) 5− x→+∞ 3 + = lim 4 x 1 x + − 2 x2 3 x2 = 5 3 31. x2 (x2 − x5 ) x2 − x5 x4 − 5x = lim = lim = lim x2 = +∞ 3 1 x→−∞ − 3x + 1 x→−∞ x2 (1 − x + x2 ) x→−∞ 1 − x3 + x12 lim x→−∞ x2 32. √ lim x→+∞ √ x2 x x2 q 1− 1 x2 q + 1+ −1+ +1 = lim x→+∞ x+3 x(1 + x3 ) q q √ √ 1 − x12 + 1 + x12 1+ 1 =2 = lim = x→+∞ 1 1 + x3 1 x2  33. lim x( p x→+∞ x2 √ p x · (x2 + 1 − x2 ) x2 + 1 + x 2 √ + 1−x) = lim x( x + 1−x)· √ = lim x→+∞ x2 + 1 + x x→+∞ x2 + 1 + x x→+∞ x q = lim x· 1+ 1 x2 1 = lim q +1 x→+∞ 3 1+ 1 x2 =√ +1 1 1 = 2 1+0+1 34. tan x z √ √ [tan x = z, x → 0 ⇔ z → 0] = lim z→0 1 − 1 + tan x 1− 1+z √ √   √ √ z 1+ 1+z z(1 + 1 + z) √ √ = lim · = lim = − lim (1+ 1 + z) = −(1+ 1 + 0) = −2 z→0 1 − z→0 z→0 1 − (1 + z) 1+z 1+ 1+z lim x→0 35. −x lim 2 x→−∞  x  −∞ 1 1 = lim = = 2∞ = ∞ x→−∞ 2 2 36. lim 3−x = x→+∞ 37. 2x 1 − 2x − 1 lim x = lim x x→+∞ 2 + 1 x→+∞ 2 1+ 38. 1 =0 3∞
1 2x
1 2x 3x+1 1 − 3x lim = lim x→+∞ 2 + 3x+1 x→+∞ 3x+1 1− x→+∞ 1 + = lim 1 3x+1 2 3x+1 1 2x 1 2x 1−0 =1 1+0 =
− 13 0 − 13 1
= =− 0 + 1 3 +1 39.  lim x→+∞ x a· xa   a x 1 1 = = lim 1+ = lim 1+ a 1+ a x→+∞ x→+∞ x x x  lim 1+ x→+∞ 1  xa !a a x 40.  lim x→+∞ x 1+x  = lim 1− x→+∞ x  = lim 1− x→+∞ 1 x+1 x+1 1 x+1 x  · lim 1− x→+∞ = lim x→+∞ 1 x+1  1− 1 x+1 x+1−1 −1 = e−1 · (1 − 0)−1 = 1 e 41.  lim x→+∞ x x+5  lim 1− x→+∞ 2x−1 = lim x→+∞ 5 x+5  1− x+5 !2 · lim x→+∞ 5 x+5  1− 2x−1  = lim x→+∞ 5 x+5 1− 5 x+5 2(x+5)−11 = −11 = (e−5 )2 ·(1−0)−11 = 1 e10 42.  lim x→+∞ 2x + 4 2x + 5  = lim 1− x→+∞ x+3  = lim 1− x→+∞ 1 2x + 5  2x+5 2 1 2x + 5  · lim 1− x→+∞ 43. 1 3 lim (1 + 3x) x = lim (1 + 3x) 3x = x→0 x→0 4 x+3  = lim 1− x→+∞ 1 2x + 5   12  2x+6 2 1 1 1 = (e−1 ) 2 · 1 2 = √ e 1 lim (1 + 3x) 3x x→0 1 2x + 5 3 = e3 = ea 44. x2 + 2x + 1 x2 + 2x + 1 x2 + 2x + 1 1 4 1 = lim = lim · lim = − · lim =? 2 x→1 x − 4x + 3 x→1 (x − 3)(x − 1) x→1 x→1 x − 1 x−3 2 x→1 x − 1 lim x2 + 2x + 1 1 = −2 · lim = −2 · (+∞) = −∞ 2 x→1+ x − 4x + 3 x→1+ x − 1 lim x2 + 2x + 1 1 = −2 · lim = −2 · (−∞) = +∞ x→1− x2 − 4x + 3 x→1− x − 1 Limita neexistuje. lim 45. x2 + 2x + 1 x2 + 2x + 1 1 1 = lim = lim (x2 +2x+1)· lim = 9· lim =? 2 x→2 x − 4x + 4 x→2 (x − 2)2 x→2 x→2 (x − 2)2 x→2 (x − 2)2 lim 1 =∞ (x − 2)2 lim x→2+ 1 =∞ x→2− (x − 2)2 lim x2 + 2x + 1 =9·∞=∞ x→2 x2 − 4x + 4 lim 46. x x 1 1 = lim · lim = 1 · lim 2 x→0 x→0 x→0 sin x sin x sin x sin x 1 lim =∞ x→0+ sin x 1 lim = −∞ x→0− sin x Limita neexistuje. lim x→0 47. x x = lim =1 x→0+ x x→0+ |x| x x lim = lim = −1 x→0− |x| x→0+ −x lim 48. 1 lim 2 x = 2+∞ = ∞ x→0+ 1 lim 2 x = 2−∞ = 0 x→0− 49.  3 1 lim x→0+ 2x + 3 1 x 3 +2 = lim x→0+ 1 x 1 2x 1 3x 3 1 x  +  1+ x→0− 2x + 3 1 x 3 +2 2x 1 3x  = lim 2 1 3x 1 lim 1 3 = x→0+ 1 3x + 1+ 3 1 3x 2 1 3x 0+3 3 2−∞ + 3 = = −∞ 3 +2 0+2 2 5 = lim x→0+ 2 3
x1 1+ 3 + 1 2 3x 1 3x = 0+0 =0 1+0